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Uniform Eberlein Compactifications of Metrizable Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 159 (2012) 1691–1694 Contents lists available at SciVerse ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Uniform Eberlein compactifications of metrizable spaces ∗ Taras Banakh a,b, , Arkady Leiderman c a Uniwersytet Jana Kochanowskiego w Kielcah, Poland b Ivan Franko National University of Lviv, Ukraine c Ben-Gurion University of the Negev, Israel article info abstract MSC: We prove that each metrizable space X (of size |X| c) has a (first countable) uniform 54D35 Eberlein compactification and each scattered metrizable space has a scattered hereditarily 54G12 paracompact compactification. Each compact scattered hereditarily paracompact space 54D30 is uniform Eberlein and belongs to the smallest class A of compact spaces, which 54D20 contains the empty set, the singleton, and is closed under producing the Alexandroff compactification of the topological sum of a family of compacta from the class A. Keywords: © Scattered space 2011 Elsevier B.V. All rights reserved. Metrizable space Scattered compactification Hereditarily paracompact space Uniform Eberlein compact space Looking for compactifications of metrizable spaces it is natural to search them among compacta having as much prop- erties of metrizable spaces as possible. In this respect, the class of Eberlein compacta fits quite well: Eberlein compacta are Fréchet–Urysohn, they contain metrizable dense Gδ-subsets, their weight coincides with their cellularity, etc. In [2] Arkhangelski˘ı showed that each metrizable space has an Eberlein compactification. Note also that the Eberlein compactifi- cation cX of X can be constructed to have dimension dim(cX) = dim(X) [4,6]. The class of Eberlein compacta contains a subclass consisting of uniform Eberlein compacta. Let us recall that a compact space K is (uniform) Eberlein if K is homeomorphic to a compact subspace of a (Hilbert) Banach space endowed with the weak topology, see [3]. The class of uniform Eberlein compact is strictly smaller than the class of Eberlein compacta [3,7]. In this paper we improve the mentioned compactification result of Arkhangelski˘ı [2] constructing a uniform Eberlein compactification for each metrizable space. Theorem 1. Each metrizable space X has a uniform Eberlein compactification. Proof. Instead of modifying the original construction of Arkhangelski˘ı [2] we shall present an alternative analytic proof. Given a metric space X, apply the classical Dowker result [5, 4.4.K] to find an embedding f : X → S to the unit sphere S of a Hilbert space H of suitable density. It is well known that the norm and weak topologies coincide on S.So,S can be considered as a subspace of the closed unit ball B of H endowed with the weak topology. In such a way we obtain an embedding of the metric space X into the metrizable subspace S of the weak unit ball B of H, which is a uniform Eberlein compactum. The closure of X in B will be the desired uniform Eberlein compactification of X. 2 Modifying the original approach of Arkhangelski˘ı, we can construct first countable uniform Eberlein compactifications. * Corresponding author at: Ivan Franko National University of Lviv, Ukraine. E-mail addresses: [email protected] (T. Banakh), [email protected] (A. Leiderman). 0166-8641/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2011.06.060 1692 T. Banakh, A. Leiderman / Topology and its Applications 159 (2012) 1691–1694 Theorem 2. A metrizable space X has a first countable uniform Eberlein compactification if and only if |X| c. Proof. The “only if ” part follows from the famous Arkhangelski˘ı’s result [5, 3.1.30] saying that each first countable compact Hausdorff space K has cardinality |K | c. To prove the “if ” part, we shall use Kowalsky’s theorem [5, 4.4.9] asserting that ω each metrizable space X of weight κ embeds into the countable power (Hκ ) of the metric hedgehog Hκ = teα: t ∈[0, 1], α ∈ κ ⊂ l2(κ) with κ spines. Here (eα)α∈κ is the standard orthonormal base of the Hilbert space l2(κ).IfX has weight c,thenX ω embeds into the countable power Hc of Hc. Since the countable product of first countable uniform Eberlein compacta is first countable and uniform Eberlein, it suffices to show that the hedgehog Hc has a first countable uniform Eberlein compactification. This compactification can be constructed explicitly. First fix an injective enumeration {ϕα: α ∈ c} of a topological copy of the Cantor set in [π/6, π/3]. The embedding f : Hc → S of the hedgehog Hc into the unit sphere S of the Hilbert space l2(c + 2) can be defined as follows: f : teα → cos(t)ec+1 + sin(t) cos(ϕα)ec + sin(ϕα)eα for t ∈[0, 1] and α ∈ c. One can show that the closure of f (Hc) in the weak topology of l2(c + 2) coincides with the set f (Hc) ∪ cos(t)ec+1 + sin(t) cos(ϕα)ec: t ∈[0, 1], α ∈ c homeomorphic to the cone over the Alexandroff duplicate of the Cantor set. The latter set is first countable and being a compact subset of (l2(c+2), weak),isuniformEberlein. 2 Remark 1. Independently and simultaneously the result that each metrizable space has a uniform Eberlein compactification has been obtained by B.A. Pasynkov [10]. Next, we shall be interested in scattered uniform Eberlein compactifications of scattered metrizable spaces. We recall that a topological space X is scattered if each subspace of X has an isolated point. According to [13] each metrizable scattered space has a compactification homeomorphic to an initial segment of ordinals endowed with the order topology. Unfortunately, such uncountable segments are not Fréchet–Urysohn and hence not Eberlein compact. Nonetheless, scattered metrizable spaces have scattered compactifications which are hereditarily paracompact and uniform Eberlein. The class of scattered hereditarily paracompact compacta is very narrow and admits a constructive description. Namely, it coincides with the smallest class A of compacta, which contains the empty set, the singleton, and the Alexandroff com- pactifications of topological sums of compacta from the class A. We recall that a compactification X of a regular locally compact space X is Alexandroff if its remainder has size |X \ X| 1. Theorem 3. 1. Each scattered metrizable space X has a scattered hereditarily paracompact compactification K . 2. Each scattered hereditarily paracompact compact space is uniform Eberlein. 3. A compact space X is scattered and hereditarily paracompact if and only if it belongs to the class A. Remark 2. The class A is not closed with respect to finite products: the product αℵ1 × αℵ0 of the Alexandroff compacti- fications of the discrete spaces of size ℵ1 and ℵ0 does not belong to the class A because it is not hereditarily normal, see [5, 2.7.16(a)]. On the other hand, αℵ1 × αℵ0 is a scattered uniform Eberlein compact space. It is known that each scattered Eberlein compact space K is strong Eberlein in the sense that it embeds into the σ -product κ (xα)α∈κ ∈{0, 1} : {α ∈ κ: xα = 0} < ℵ0 of the two-point space for some cardinal κ, see [1]. Remark 3. In light of Theorem 3 it should be mentioned that there are scattered countable regular (and hence hereditarily paracompact) spaces admitting no scattered compactification, see [8,9,11,12]. The remaining part of the paper is devoted to the proof of Theorem 3, which is done by transfinite induction using two ordinal functions: the scattered height of a scattered space and the complexity number of a space K ∈ A. (α) = (β) (0) = The scattered height of a scattered space X is defined using the α-th derived sets X β<α(X ) where X X and A denotes the set of non-isolated points of a space A. It is easy to see that a space X is scattered if and only if X(α) is empty for some ordinal α. The smallest ordinal α such that |X(α)| 1iscalledthescattered height of X and is denoted by ht(X). For example, a space X with a unique non-isolated point has ht(X) = 1. T. Banakh, A. Leiderman / Topology and its Applications 159 (2012) 1691–1694 1693 A A = A A Next,weobservethattheclass can be written as the union α α where 0 is the class of all spaces X with |X| 1, and Aα consists of the Alexandroff compactifications of topological sums of compacta from the class A<α = A ∈ A ∈ A β<α β .Thecomplexity number cx(K ) of a space K is the smallest ordinal α such that K α. Lemma 1. Any compact space K ∈ A is a scattered, hereditarily paracompact, uniform Eberlein and has cx(K ) = ht(K ). Proof. This lemma will be proved by induction on cx(K ). The assertion is trivial if cx(K ) = 0 (in which case |K | 1). Assume that the lemma is proved for compacta X ∈ A with cx(X)<α. Fix any compact space K ∈ A with cx(K ) = α > 0. { ∈ I}⊂A Then there is a family Ki : i <α such that K is the Alexandroff compactification of the topological sum i∈I Ki . By the inductive assumption, each space Ki is scattered, hereditarily paracompact, uniform Eberlein and has cx(Ki ) = ht(Ki ). Then cx(K ) = sup cx(Ki) + 1 = sup ht(Ki) + 1 = ht(K ). i∈I i∈I To show that K is scattered and hereditarily paracompact, take any infinite subspace X ⊂ K . Being infinite, the set X meets some set Ki .SinceKi is scattered, the intersection X ∩ Ki contains an isolated point which is isolated in X too ∩ (because X Ki is open in X).
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